The invention relates to a method for determining the relative position, velocity, and/or acceleration of a body displaceable in a one to three dimensional space, wherein a plurality of linear acceleration sensors, which in each case have a sensitive measurement axis, is arranged and wherein the individual linear acceleration sensors are always arranged on a position Pi which is stationary with respect to the body. Furthermore, the invention relates to a device for determining the relative position, velocity, and/or acceleration of a body in a one to three dimensional space, wherein the device has a plurality of linear acceleration sensors stationarily arranged with respect to the body for capturing at least one acceleration measurement signal {right arrow over (a)}=(a1, a2, a3, a4, . . . an).
A relative position is to be understood as a position relative to a reference position, for example, a starting position. A linear acceleration sensor is to be understood as a unidirectional acceleration sensor, which is sensitive to accelerations that have at least one component that lies on the measurement axis of said linear acceleration sensor. The linear acceleration sensors can be structurally separated from each other, each one being arranged in its own housing. However, it is also possible for at least two and especially three linear acceleration accelerators having different measurement axes to be integrated as a multidirectional acceleration sensor in a common electric or electronic component.
A method and a device for determining the angular velocity of a turnable body, such as a vehicle, are disclosed in DE 199 62 687 A1. For measuring an acceleration measurement signal, a total of nine linear acceleration sensors, in each case having a sensitive measurement axis passing through the corresponding position, are fix mounted on the body in four non-adjacent positions. Three linear acceleration sensors, which in each case are oriented with their sensitive measurement axis toward one of the x, y, z axes of the body-fixed coordinate system, are provided in a first position arranged in the origin of the body-fixed coordinate system. Two other linear sensors, of which one is oriented with its measurement axis in z-direction and the other of which is oriented with its measurement axis in x-direction, are provided in a second position arranged at a distance r1 from the origin on the y-axis. Two other linear sensors, of which one is oriented with its measurement axis in y-direction and the other of which is oriented with its measurement axis in z-direction, are provided in a third position arranged at a distance r2 from the origin on the x-axis. In a corresponding manner, two linear sensors, which are oriented in x- or y-direction, are provided in a fourth position arranged on the z-axis at a distance r3 from the origin. The acceleration {right arrow over (a)} at a given point P is expressed as:{right arrow over (a)}=A(t)+{right arrow over ({dot over (ω)}x{right arrow over (r)}+{right arrow over (ω)}x({right arrow over (ω)}x{right arrow over (r)})+2·{right arrow over (ω)}x{right arrow over ({dot over (r)}+{right arrow over ({umlaut over (r)}  Equation (1)
Where:
A is the acceleration of the origin of the body-fixed coordinate system,
t is the time,
{right arrow over (ω)} is the angular velocity of the body, and
{right arrow over (r)} is the position vector that indicates the point P from the origin of the body-fixed coordinate system.
Assuming that a linear acceleration sensor is fix mounted on the point P of the body, the terms {right arrow over ({dot over (r)} and {right arrow over ({umlaut over (r)} become equal to zero. With this assumption, one obtains:{right arrow over (a)}=A(t)+{right arrow over ({dot over (ω)}x{right arrow over (r)}+{right arrow over (ω)}x({right arrow over (ω)}x{right arrow over (r)})  Equation (2)
This is a three dimensional, non-linear differential equation system, which as a rule cannot be solved analytically. The determination of the angular velocity of the turnable body therefore requires a relative complicated calculation. The device operating according to the procedure is therefore correspondingly complicated, expensive, and this notwithstanding, usually inaccurate.
DE 199 62 687 A1 discloses another prior art procedure in which the angular acceleration {right arrow over (ω)}x of the body about the x-axis is measured by only four linear acceleration sensors fix mounted on the body. In this arrangement, two linear acceleration sensors are arranged in the origin of the body-fixed coordinate system, wherein a first linear acceleration sensor is oriented with its measurement axis toward the y-axis and a second linear acceleration sensor is oriented with its measurement axis toward the z-axis. A third linear acceleration sensor is arranged on the y-axis at a distance r1 from the origin and oriented toward the z-axis. A fourth linear acceleration sensor is arranged on the z-axis at a distance r3 from the origin with its sensitive measurement axis oriented toward the y-axis. If one plugs the positions of the four linear acceleration sensors into equation (2), algebraic transformation gives:
                                          ω            .                    x                =                                                            a                                  z                  ⁢                                                                          ⁢                  1                                            -                              a                                  z                  ⁢                                                                          ⁢                  0                                                                    2              ·                              r                1                                              -                                                    a                                  y                  ⁢                                                                          ⁢                  3                                            -                              a                                  y                  ⁢                                                                          ⁢                  0                                                                    2              ·                              r                3                                                                        Equation        ⁢                                  ⁢                  (          3          )                    
wherein az0, az1, ay0, ay1 represent the measurement values of the four linear acceleration sensors. Equation (3), however, only applies to the special case where the first and third linear acceleration sensors are always oriented with their measurement axes exactly in y-direction and where both of the other linear acceleration sensors are oriented with their measurement axes perpendicular thereto in z-direction. The perpendicular arrangement of the linear acceleration sensors, however, is problematic in practice, as manufacturing and positioning tolerances arise in the manufacture and attachment of the linear acceleration sensors on the body. Deviations in positioning and alignment are almost inevitable, especially when mounting the linear acceleration sensors on the body. Even small deviations of the position of the linear acceleration sensors from the sensor arrangement on which equation (3) is based can lead to relatively large errors in the calculation of the angular acceleration {right arrow over (ω)}x. This is especially disadvantageous when the angular acceleration signal for determining the position of the body integrates and the errors add up without limit. This last point in particular was the decisive reason why the use of position determining devices that employ linear sensors only never caught on.
It is therefore the object of the invention to develop a method and a device as mentioned in the introduction that enable a simple, exact determination of the position, velocity, and/or acceleration of one of the bodies.